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The Mean Value and Rolle's Theorem
Daniel Shak The two theorems are used in calculus to allow mathematicians to make generalizations about the derivatives of a graph. They are also used to provide proof/evidence of why a graph behaves in a certain way. The Mean Value Theorem and Rolle's Theorem are often required on the AP test to prove an answer. Rolle's Theorem - Defines when there must be a derivative equal to 0 (the slope of the tangent line is 0) Graph has to satisfy three points *the graph is continuous on the closed interval a,b *the graph is differentiable on the open interval (a,b) *f (a) = f (b) -When the three requirements are met there should be a point © on the graph between a and b where the derivative equals 0. [ f ' ©=0] Mean Value Theorem - Guarantees that there exists a number on an interval with a specific derivative/slope Graph must follow- *is continuous on the colsed interval a,b * is differentiable on the open interval (a,b) - then there is a number © between points a and b where the derivative of c is equal to the slope of the secant line through a and b. :: - f ' © = [ f (b)- f (a) ] / b-a Example 1 Find the value of c that follows Rolle's Theorem for, : f(x) = 5 - 12x + 3x^2 1,3 : - The graph is continuous and differentiable from (1,3) : - f ( 1) = -4 f ( 3 )= -4 : - f ( x) satisfies Rolle's theorem so there must be a c between 1 and 3 where f ' ©=0 : - f ' (2)=0, when x=2 the derivative equals 0 Example 2 From the graph of f, estimate the values of c that satisfies the conclusion of the mean value theorem for the interval [ -5,5 ]. : - We can see that the graph is continuous and differentiable on the interval [ -5,5] : - slope of secant line f (-5) - f (5)/-5-5 = 3/5 : - appproximate values of c - x= -3.2, -0.6, 1.8, 4.1. At these points the slope is 3/5 Example 3 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then estimate all numbers c that fit the conclusions of the Mean Value Theorem. : - f(x)=x^3+x-1 -2,2 : -The graph of f is continuous and differentiable on the interval -2,2 : - slope of secant line f(2) - f(-2)/ 2- (-2) = 9/2 : - f ' ( -1 ) ~ 9/2 : - f ' ( 1 ) ~ 9/2 : - c = -1 ,1 Mini Quiz Use the graph to (a) estimate the values of c for Rolle's Theorem (interval -2,2) and The Mean Value Theorem (interval 0,2). (b) Explain why you can use Rolle's Theorem and The Mean Value Theorem to estimate the derivative at a certain point on the graph. What are the restrictions when using the two theorems?